Motivation

1. Motivation • Find a potential technological improvement to increase propulsive efficiency and thrust-weight ratio. • One possible way is to consider burning fuel in the turbine passage. Since power is extracted from the turbine, the flow temperature is lowered so fuel can be added in the turbine passage and burnt without surpassing the maximum temperature limited by the material of the turbine blade.

2. Objectives • Our goal is to study the flow physics of turbulent combustion in the turbine passage. The long term goal is to perform Large Eddy Simulation. • The current approach is to simulate the flow environment of the turbine passage by performing numerical computations of unsteady, accelerating, and chemically reacting mixing layers. • Develop a robust and efficient computer code to perform numerical computations.

3. Governing Equations • The governing equations consist of the 2-D Compressible Navier-Stokes equations, the species equations, and the turbulence equations. • For simplicity, a one-step chemical reaction mechanism is used. • A k- turbulence model is employed.

4. Approaches • The numerical computations of the flow in the turbine passage is complex due to the 3-D geometry of the turbine blade. As a result, the geometry has been simplified to straight and curved ducts that impose axial and transverse pressure gradients to the flow. • Explicit finite volume scheme was used in the past. It was robust but the CFL number was strictly limited. • Implicit finite difference is being developed. The CFL number is significantly larger than the explicit scheme.

5. Schematic of the ducts

6. Numerical Method I:Explicit scheme • Finite volume scheme is used to solve the integral form of the governing equations. The scheme is second order accurate in space. • In viscous calculations, very fine grids are required to resolve the thin mixing layer. Stiffness arises and reduces the size of the time step significantly. • Multigrid scheme is implemented to accelerate convergence to steady state solution.

7. Numerical Method I:Explicit scheme • Chemical reaction introduces another stiffness problem due to fast reaction rate. Very small time step has to be used in order to capture the chemistry accurately. • Operator-Splitting is implemented to relieve the stiffness.

8. Multigrid method • In multigrid method, several grid levels are used. Typically, the grid spacing is doubled going from finer grid to coarser grid. • The multigrid cycle starts at the finest grid. The flow variables and residuals are interpolated to the coarse grid. The variables at the coarse grid are updated and the newly obtained residuals are interpolated to the next coarser grid. This process repeats until the coarsest grid is reached. The corrections are then passed back to the next finer grid by bilinear interpolation.

11. Results I

12. Numerical Method II:Implicit scheme • A second order implicit finite difference is being developed. It allows significantly larger time step that relaxes the stiffness in viscous calculations. • Finite difference method is chosen because it is relatively easier to implement higher order scheme as required in LES in the future. • The scheme is not fully implicit because the viscous terms that contain cross derivatives are treated explicitly. • Extra computational work is required for the matrix inversion and evaluation of the Jacobians.

13. Results II

14. Future work • Perform unsteady calculations by using the implicit scheme. • Implement Operator-Splitting in the implicit scheme.